Dynamic covariate balancing: estimating treatment effects over time with potential local projections

Abstract

This paper studies the estimation and inference of treatment effects in panel data settings when treatments change dynamically over time. We propose a balancing method that allows for (i) treatments to be assigned dynamically over time based on high-dimensional covariates, past outcomes, and treatments; (ii) outcomes and time-varying covariates to depend on the trajectory of all past treatments; (iii) heterogeneity of treatment effects. Our approach recursively projects potential outcomes' expectations on past histories. It then controls the bias arising from the non-experimental and sequential nature of this setting by balancing dynamically observable characteristics over time. We establish inferential guarantees of the proposed method even when the number of observable characteristics significantly exceeds the sample size. We study numerical properties of the estimator and illustrate the benefits of the procedure in an empirical application.

Figures (10)

• Figure 1: Sampling process in two periods. First, baseline covariates $X_{i,1}$ realize at $t = 0$. Then, treatment $D_{i,1}$ is assigned and the outcomes and covariates $(Y_{i,1},X_{i,2})$ realize at $t = 1$. Finally, the treatment $D_{i,2}$ is assigned and, afterwards, the endline outcome $Y_{i,2}$ realizes.
• Figure 2: The left panel illustrates all the possible causal paths under Sequential Ignorability (Assumption \ref{['ass:seqign']}). Here, past treatments may affect intermediate covariates, and future treatments may depend on past treatments, covariates and outcomes. The right panel presents two estimands of interest. In particular, $\mathrm{ATE}(\mathbf{1}, \mathbf{0})$ (the effect of increasing treatments in both periods) denotes the effect mediated through all red edges, including the dotted red edge. Instead, $\mathrm{ATE}((1, 0), (0, 0))$ (the direct effect of only increasing treatment in the first period) denotes the effect mediated through all red edges excluding the dotted red edge.
• Figure 3: The figure illustrates the dynamics of treatments.
• Figure 4: Estimated probability of treatment for one year (left-panel) and two consecutive years (right-panel). Estimation is performed via logistic regression with a pooled regression with year, region fixed effects and four lagged outcomes. The right panel also controls for the past treatment assignment. The figure illustrates the sensitivity of inverse probability weights to longer time horizon, motivating more stable balancing weights.
• Figure 5: Left-hand side: pooled regression from $t \in \{1989, \cdots, 2010\}$. Gray region denotes the $90\%$ confidence band for the least parsimonious model. DCB and DCB2 refer to two separate specification, with DCB corresponding to the more parsimonious one. LP denotes a local projection on $t$ periods before. (no fe) indicates a specification without country fixed effects. Right-hand side reports $\log(|I(dcb)| + 1)/\log(|I(aipw)| + 1) - 1 \approx |I(dcb)|/|I(aipw)| - 1$, where $I(\cdot)$ denotes the imbalance (as in Lemma \ref{['lem:balancing1']}) in the lagged outcome usin DCB or IPW weights.

Theorems & Definitions (35)

• Example 3.1: Observed outcomes
• Lemma 3.1: Identification
• Example 3.2: Linear Model
• Remark 1: Linearity in high-dimensions as an approximation to the true model
• Remark 2: Comparison with standard local projections and DiD
• Theorem 4.1: Balancing weights
• Remark 3: Estimating when-to-treat policies
• Lemma 4.2
• Theorem 5.1: Existence of feasible weights
• Corollary 1